You might want to look at alternative solutions.
First, dropbox is a decent way to share files, but at best would be for one person sharing across multiple locations. I don't even know where to begin with the conflicts that you're going to run into trying to use dropbox as your source management system. Look into git or svn. svn is pretty user friendly with tortoise on windows, and there are places that offer free repos.
As for mysql, trust people here when they say you don't want to do this. You will trash your data, and quickly put it in an inoperable state. Since your using dropbox, reverting back to a operable state will be extremely difficult or impossible, depending on how well you did backups and timed them.
Your going to host the website somewhere, no? Don't they provide a mysql db for you to use? If so, you should check and see if you can expose ports for remote managing it, or just manage it through the hosts phpmySQL interface. Unless your making structure changes, you probably don't need to be in deep with the database anyways. You say that you have the DB on your machine but your friend can't access it, maybe you can open a port for him?
Finally, even if you did get the files to sync without trashing your data, he's still going to have to setup a MySQL instance on his side to do something with the datafiles. MySQL consists of two parts, the software and the data, you can't just dump the data into dropbox (or any sharing method) and have it be available to someone else who has access to the datafiles. If I'm mistaken and he does have mysql server configured, why can't he just install his own set of tables?
3What database are you using for WordPress? Unless you use something like a SQLite database, you will not be able to sync it. MySQL will never sync. – TFM – 2013-03-16T13:10:16.917
Are you talking about whole WordPress installation or backed up archived databases? I know that if you want others to collaborate, you would have to invite them by typing in their emails. This is done through Dropbox web interface. – Boris_yo – 2013-03-16T13:21:56.393
@TFM MySQL, I would like to have my local databases on Dropbox together with all the other Wordpress files – user1301428 – 2013-03-16T21:12:14.533
@Boris_yo I would like to have the whole Wordpress installation on Dropbox, databases included – user1301428 – 2013-03-16T21:12:55.983
1Sorry, but this is not a good idea and is just not going to work the way you want it to. In particular, I'm afraid it's quite impossible with MySQL. You'd be better served setting up a server or a cheap shared web hosting account somewhere to host your site in development (or at least the MySQL server). – tgies – 2013-03-20T07:46:49.607
@tgies I didn't think that changing the path to my MySQL databases would be so difficult... – user1301428 – 2013-03-20T09:52:55.617
1@user1301428 That's not the difficult part. The issue is that it's a Dropbox folder. Dropbox is going to go absolutely crazy and thrash around trying to keep the file in sync, and if you ever try to access the database at the same time as the other user, you're going to have concurrency issues. – tgies – 2013-03-20T13:50:16.233